Suppose God offers you the following choice.
(1) He will actualize you in world W where you will be in hell along with infinitely many others. The infinite set of persons P in hell is mapped onto the set of naturals; whatever number n that is assigned to you is the number of minutes you spend in hell. You then go permanently to heaven.
(2) He will actualize you in W’ where you will be in heaven along with infinitely many others. The set of persons P in heaven is mapped onto the set of naturals; whatever number n that is assigned to you is the number of miniutes you will spend in heaven. You then go to hell permanently.
Suppose you try determine what to do by taking the point of view those who are very bad off. As it happens, for any P in W whose point of view you take, you are better off being P in W than being any person P’ in W’. For any person P in W, the expected value of being in W is equal to the badness of a terminating period of time in hell + the goodness of an endless period of time in heaven. For any person P’ in W’, the expected value of being in W’ equals the goodness of a terminating period of time in heaven + the badness of an endless period of time in hell. So, every person in W is better off than anyone in W’. On the other hand, W will always have an infinite number of people in hell (and a finite number in heaven) and W’ will always have an infinite number of people in heaven (and a finite number of people in hell). So the overall value of W is lower than the overall value of W’, yet the expected value of every person in W is much higher than the expected value of anyone in W’. Clearly, something has gone wrong in assessing the overall value of worlds, since any rational and impartial person would prefer W to W’, and yet we have claimed that W’ is overall better than W.
What’s gone wrong, I say, is the following inference: there is permanently an infinite number of people in hell in W and there is permanently an infinite number of people in heaven in W’, therefore W is overall worse than W’. Not so! Every rational person would prefer being in W (no matter what natural number they are assigned) to being in W’ since everyone in W is better off than anyone in W’.
Relevant Prosblogion discussions:
Hi Mike,
Interesting post! If the overall value of a world is the sum of the basic intrinsic desires of individuals in the world do you get the result that W is better than W'? You have to screen off an infinite amount of suffering in W, but, as you say, for each individual in W, it's better for them to be in W than W'. This should come out in terms of their basic intrinsic desires, i.e., each s in W desires to be in W rather than W'. I'm not exactly sure how basic intrinsic desires are explicated. If you just look at it as a utilitarian that doesn't make cross-world comparisions then W is worse off than W' for the reason you give; but, if each individual is better off in W, that should come out in the overall value in the way I mentioned--too bad for simple utilitarianism.
Mike,
"... W will always have an infinite number of people in hell (and a finite number in heaven) and W’ will always have an infinite number of people in heaven (and a finite number of people in hell). So the overall value of W is lower than the overall value of W’ ..."
1. Q. Smith, in his 2003 argument for moral nihilism discussed in your last post, focuses on the total value of the WHOLE universe-HISTORY, not on the total value of the universe at A TIME. (Cf., e.g., p. 45: "Each time has at least one unit of value. Since there are an infinite number of times, say hours, ordered in the order of the positive integers, omega, the A-history and B-history both have the same number of units of value, aleph-zero.")
2. I guess he'd say that both histories of W and W' have these features: they contain
infinity of humans,
infinity of beatific human states (minutes),
infinity of hellish human states.
But "... all organisms, inanimate mass and energy, and space and time, and states of these entities, have value ..." (p. 44).
Every human has at least one unit of positive value. Every human beatific state (minute), too. Every human hellish state has at least one unit of negative value.
Thus, both W and W' contain infinitely positive and infinitely negative value.
3. What about the OVERALL value of W and W'?
Isn‘t the overall value of any entity captured by the sum of its positive value minus the sum of its negative value?
If so, isn’t in Cantorian transfinite arithmetic, likely embraced by QS in his paper, both subtracting from positive transfinite numbers and adding to negative transfinite numbers disallowed/not defined?
If so, however, then the total (overall) value of the world W is not allowed/defined (in Cantorian arithmetic). Because what else would be the total value of the world W (or W') than the sum of its positive value minus the sum of its negative value?
If you allow subtracting from positive transfinite numbers or adding to negative transfinite numbers, what is the overall value of W (or W')? Zero? That would seem like a natural answer.
4. Finally, and little offtopic, I want to ask you, Mike, again: You wrote in the discussion to your post: „Yes, I think I have two options, deny transitivity for equality, which cannot rationally be done, or hold that the loss of a penny is a loss of 1/ooth of my total wealth.“ I believe oo/1 is not allowed in Cantorian transfinite arithmetic. Is 1/oo allowed? What does 1/oo equal?
Ted,
Thanks. Yes, I think if we take a preference-satisfaction view of value, then W has to come out better than W', since the considered preferences of everyone in W are better satisfied than the preferences of those in W'. This need not be a subjectivist account of value either, since we could require that only preferences for objectively valuable things count in the assessment of individual welfare, and so in the assessment of the overall value of worlds. That's a nice suggestion!
Vlas,
I don't see off hand how this observation is relevant to what I say in the post.
Q. Smith, in his 2003 argument for moral nihilism discussed in your last post, focuses on the total value of the WHOLE universe-HISTORY, not on the total value of the universe at A TIME.
But you add,
. . .the total (overall) value of the world W is not allowed/defined (in Cantorian arithmetic). Because what else would be the total value of the world W (or W') than the sum of its positive value minus the sum of its negative value?
I'm not pretending that addition/subtraction is well-defined for Cantorian infinities. Indeed I'm counting on it being true that oo - 1 (or + 1)= oo. This is supposed to account for why W is arguably worse than W'.
Is 1/oo allowed? What does 1/oo equal?
No, it isn't. But it is allowed in non-standard accounts according to which it is an infinitessimal.
Mike,
Thanks for the reply.
1. How is my comment relevant?
Again, you wrote:
"... W will always have an infinite number of people in hell (and a finite number in heaven) and W’ will always have an infinite number of people in heaven (and a finite number of people in hell). So the overall value of W is lower than the overall value of W’ ..."
When assessing the total value, it's natural to focus on the total value of the WHOLE universe-HISTORY, not on the total value of the universe at A TIME.
Both histories of W and W' contain:
infinity of humans,
infinity of beatific human states (minutes),
infinity of hellish human states.
Every human has at least one unit of positive value. Every human beatific state (minute), too. Every human hellish state has at least one unit of negative value.
Thus, both W and W' contain infinitely positive and infinitely negative value.
If you allow subtracting from positive transfinite numbers or adding to negative transfinite numbers, what is the overall value of W and W'? Zero? That would seem like a natural answer in both cases. But then it is not true that the overall value of W is lower than the overall value of W’. The overall values are equal because they both equal zero.
2. I said: IF subtracting from positive transfinite numbers or adding to negative transfinite numbers is disallowed/not defined, then the overall value of W and W' is not allowed/defined.
Addendum
3. I use above "defined" ("not defined") in the same sense as (syntactically) "allowed".
In Cantorian arithmetic, I can say "oo + 1" or "oo + 1 = oo". It is allowed to say that. And oo + 1 has a determined, that is, defined, value: it's oo.
Not so for "oo - 1", "oo/1", "oo - 1 = oo" or "oo/1 = oo". Subtraction and division is disallowed. And oo - 1 and oo/1 are not defined. (Cf. W. L. Craig, The Kalaam Cosmological argument, 1979, pp. 80ff.)
But even in Cantorian arithmetic, oo + 1 is not "well-defined" in Mike's sense: the determined value of oo + 1 is not different from oo.
Am I right, Mike?
When assessing the total value, it's natural to focus on the total value of the WHOLE universe-HISTORY, not on the total value of the universe at A TIME.
By hypothesis, W will always have an infinite number of people is hell and W' will always have an infinite number of people in heaven. So I'm not talking about a time-slice of either world. The finite addtions/subtractions are not going to be relevant, at least not on Smith-style assessments of value for worlds. This is part of what is wrong with Smith's assessment. He has W coming out worse than W', when in fact it is better.
Mike,
Thanks.
I still don't see how QS has W coming out worse than W'. If he allows the overall value of W and the overall value of W', then it rather seems that he has them as equal.
Great argument (as is the case in Jon's post). I wonder if one could use considerations of this sort to construct an argument against presentism. If one has an eternalist view, it is very natural to say that W is better than W' because we should look at people's lives as a whole. But on a presentist view, it is very natural to take the momentary judgments, rather than the whole-life judgments, as more basic, and then one will get W' better than W.
Another fun thing about this case. Suppose you don't believe in persistence over time: You think there are only person-stages, and that tomorrow (or in a second) someone else will be in your place. Then W is not in any way better than W'. It is only better if there is persistence over time that we can make the distinction between W and W'. But plainly we should make such a distinction. Therefore, the case gives a nice argument for the significance of persistence over time to ethics.
The case is also a nice illustration of why it is that in two-variable calculus the order of integration in general matters.